Oscillation Theorems for Linear Second Order Ordinary Differential Equations

Abstract

We are here concerned with the oscillatory behavior of solutions of the equation (1) x" + a(t)x = 0, where a(t) is locally integrable on [0, oo ). Our main result is an extension of a nonoscillation theorem due to Hartman [2, Theorem II], the contrapositive of which is a useful criterion for equation (1) Simple examples of averaging pairs are (1, 1), ((f+1)-1, t + 1), ((¿ + 1)~1/2, log (/ + 1)). In fact, any pair (<r, a), with (tr, a) satisfying (i), o--a bounded, and f£ ais)ds = oo, is an averaging pair.Presented to the Society, June 17, 1967 under the title An oscillation criterion for linear second order ordinary differential equations;

A Second Order Superlinear Oscillation Criterion

“…It is noteworthy that the conditions (A x ) and (A 2 ) are also sufficient for the oscillation of the differential equation (E) with /(y) = y, y e R. The validity of this result for the linear case follows from a result of Hartman [2], see also Coles [1] and Macki and Wong [4]. Also, the well-known criterion of Wintner [7] I then the condition (A 2 ) implies oscillation even in the case where (A x ) fails (cf.…”